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Using the dupire method: $$\sigma(T, K)=\sqrt{\frac{\sigma_{\mathrm{imp}}^2+2 \sigma_{\mathrm{imp}} T\left(\frac{\partial \sigma_{\mathrm{imp}}}{\partial T}+(r-q) K \frac{\partial \sigma_{\mathrm{imp}}}{\partial K}\right)}{1+2 d_1 K \sqrt{T} \frac{\partial \sigma_{\mathrm{imp}}}{\partial K}+K^2 T\left(d_1 d_2\left(\frac{\partial \sigma_{\mathrm{imp}}}{\partial K}\right)^2+\sigma_{\mathrm{imp}} \frac{\partial^2 \sigma_{\mathrm{imp}}}{\partial K^2}\right)}}$$

According to Clark (2011) we're only supposed to interpolate across the time domain because, there is no connection with vol across different strikes. For stock options, do I interpolate the implied vol surface across both strike and time?

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